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Analysis Simulation of Interaction Information in Chaotic Systems of Fractional Order
Here it is important to note that at the description of
properties of systems with fractional structure it is impossible
to use representation of Euclidean geometry.
There is a need of the analysis of these processes for terms
of geometry of fractional dimension.
Remark. In  presented results of communication of a
fractional integro-differentiation (in Riman-Liouville or
Gryunvalda-Letnikov’s terms) with Koch’s curves.
It is noted that biunique communication between fractals
and fractional operators does not exist: fractals can be
generated and described without use of fractional operations,
and defined the fractional operator not necessarily generates
defined (unambiguously with it
connected) fractal process or fractal variety.
However use of fractional operations allows generating
other fractal process (variety) which fractal dimension is
connected
with an
indicator
of a
fractional
integro-differentiation a linear ratio on the basis of the set
fractal process (variety).
In  fractional integrals of Riman-Liouville are
understood as integrals on space of fractional dimension.
Thus the indicator of integration is connected with dimension
of space an unambiguous ratio.
In this regard consideration of dimension of chaotic
systems of a fractional order causes interest. So, in  was
noted that dimension of such systems can be defined by the
sum of fractional exponents Σ , and Σ  3 is the most
effective.
Let the chaotic fractional system of Lorentz take place :
d
d
d
(3)
x    y  x
y  x  y  xz'
z  xy  bz .

dt

dt

dt

Let the trajectory of the generalized memory system is of
the form [6-8]:
(5)
QGM = Q≥0 Q≤0 .

Definition 2. Q≥0 ⊂QGM - it called semi-trajectory to
the trajectory QGM , if each t &gt; 0 it the inclusion
j 1

Q≥0  Omem  t j , t j 1 , j  .

trajectory

Here Q≤0 sk , sk 1  - segment of the semi-trajectory “loss
memory” of responsible values t sk , sk 1  , and O - ε neighborhood of the corresponding set.
Definition 4. Semi-trajectory Q≥0 ⊂Q recurrent if for
lm

inclusion

Q≥0 ⊂O  t j , t j 1 , j .
j -1

j 0

(8)

Here Q≥0 ⊂ t j , t j 1 - segment of the semi-trajectory

[

]

recurrent of responsible values t ⊂ t j ,t j +1 .
Definition 5. Semi-trajectory Q≤0 ⊂Q not recurrent if for
every
it
the
inclusion
t&lt;0
k

Q≤0 ⊂ O   sk , sk 1 , k  1 .

(9)

k 1

Remark. It is known the average return time of Poincare is
determined by the fractal dimensions the trajectories of
generalized memory, GM. Hence the memory loss will be
determined by the difference between global and local fractal
dimensions, which means that the recurrence and
no-recurrence semi-trajectories respectively.
C. Formation of Loss Memory
It is a known that during the Poincare recurrence
characterizes as a “residual”, and the real memory of the
fractional-order system. Hence the equivalence between the
spectrums of the Poincare returns time and distribution of
generalized memory.
Los memory is determined by the difference between the
global and the local fractal dimensions, which means,
respectively, reversible and irreversible processes. Loss of
information numerically defines the entropy.

W at an angle of
communications of average time of return of Poincare 
d f and “residual” memory J t  .
Here g :   d l : d  J t  :    g ,l  .
From here U   X ,   - the generalized compact metric

with

D. Spectrum dimensions of the Poincare Return Time
From the perspective of mathematics synergetic aspect of
the process of evolution is a change of a topological structure
of the phase space of an open system.
Tracking change this structure as transient process requires
the formation of a generalized criterion for recognizing the
system in a certain state of chaos-quasi-periodic-hyper chaos
and so on.

f

t&gt;0

every

Let’s consider transformation

space of Poincare with dimension

(7)

k 1

f

f

ε -

QGM , if each t &lt; 0 it the inclusion
k

compact fractional metric space with

and Omem -

Q0 ⊂Olm  sk , sk 1 ,k  1 .

dimension  .
Let’s designate W  X , d  . On the basis  and Remarks
~ 
X , A ,  W .

]

neighborhood of the corresponding set.
Definition 3. Q≤0 ⊂QGM it called semi-trajectory to the

~
X - any set
~

of nonlinear physical systems, A - a subset of a set X of
~

systems of a fractional order with memory A  X . Then a

[

of responsible values t ⊂ t j ,t j +1

This, in the context of fractional dynamics let

X~ , A ,  –

(6)

Here Q≥0 t j , t j 1 - segment of the semi-trajectory memory

Here   10,   28, b  8 / 3; 0   ,  ,  1, r  1.
Then fractional dimension of system of the equations (3)
will have an appearance :
(4)
      .
So, for example, for Lorentz’s system with fractional
exponents       0.99 , effective dimension
  2.97 .

j 0

.

B. Generalized Memory
As said, the analysis and synthesis of multidimensional
chaotic system of fractional-order there was a problem with
memory estimation.
Axiomatic

86

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